Cone Rigidity Them .

( M"

, g . pl : complete . Rc 7 - in-DS, I B , 1pm z v so ,

I Bglpsl-

lBib e s.

- s v.Is 1lb )

then day I B. Ipl , B. Iz'T ) a ¥ Is In -v )

.

where CIZ) is a metric cone w/ vertex z't

.

" IZip Zz I Ey . pyz.

p# "

Yi Y' angle of comparison o in R2.

"

We say h is a S - conical function on Brlp ) it

• f 18h - zgl'

s s

Brlpl

• f 16h12- 4h I s Sr'

Brys)

• f I ylh - dist s Sr dplxl = dip . x ) .

Brlp)

• sup Ih - dial a srz

Brlp)

tem Suppose Rc z - in -118,

h is a S - conical function

on 13444 . For any E > o, lpxl = E

,& V-te-o.IT .

F y et .

I lpxltlxyl - lpyl I + I lxyl - tl c Ils In )/s .

x t

P

SM : spherical bundle → M.

dm : Liouville measure on SM.

step, l ' 0h - v > -"tutti ) - Huo, ,I t t / dah )

E t . f low - zgl a c.rs.

1341ps

§zµ, I 01h - dp'll < ants.

3- I . I v e Sy M ,sit. 1×51 c Ils In )

I v - Odp III / s Els In),

& dp is smooth at I .

y

( ohhh - zdplxyv I + / ( oh .us - hl④IhHk + tf-

< I.

[ §, ,p ,t c T .

V x t Byzlpl , ht ft lo .E ) .3- It Be Ix) , st.

f- txt a Ekin ) . ]

t = IIYI .

I k - dit c CS on 13,47 . y = Kiel .

I Ctpxltts' - lpytl = / dptxitztdplxltt' - dishes, I

← I hunt ztdplxl"'t't ' - hunt'll t E

.

V 2 Odp III . zdptxlv I Pdp'

II) = oh let -

E I hlkloil-hlkltil-tZ-tahlxl.us/< I

.

I 1pm + t - lpyl / c4-

e Els .

lpxnltttlpyl

Ixyl 7 lpyl - lpxl z lpyl - lpxT - I 7 t - Hs.

lxyl ± lxnyl + I ← LEK Ito . to ] t t

e t -1 I.

I Kyl - t I < Else .

( Mj , g; , p; ): complete

Rc Eg;] z - (n - t) of ,I 13,4311 Z V s o .

( X , dx , p ) = YI. ( Mj , g ; . p;) .

( Y . dy ,

o ) is atat x E X if

F ra → o ( a → as ),sit .

( Y , dy, ol = him ( X , ri'd , . x ) .

x -so

RLX) = { x e X '- ⑦ tangent cone at x = R

" }.

regular point .

• Every tangent cone of X is a metric cone

w/ diam E Tu. length space .

→cross - section

PI ( Y , dy .

o ) = him ( X , ri'd × . x )* → o

ta - s o.

God : ht R > o . Belo) ± Bp Iz 't ) ,CIZ) : metric cone

diam Z E T

z't

: vertex .t s > o

'

=dealt

d ( Belo) , BRIX 's ti'd x ) ) - E, for large a

.

X; t Mj , x;→ x

dat ( Bp, lxidx ) , Bp, ( x; ) ) s EI

for large j ( depending on a ) .

dah l Belo ) . ( Bp Ix; ; rig; ) . rig; ) ) e 2L.

-

t.ir ) =

/ Brigitx; → x .

f - Z Col n , v . lpxl ) .VI. Ir )

for r E I .

I = { SR ra E ( o , I ] : look ra c l , se Q . as I } .

is countable.

By taking a diagonal . we may assume

I flr ) = Yin f; Irl , for re D .

1-

For large p , rp car. .

'¥+150 Rrp ) - f- ( 100 Rrp) c E .

*

÷.

f- 15h - ' Rr. ) - flz-kkr.IE thin f- In - f- Irr. )+→ o

For large j ( depending on p )

f ; I sokrp ) - f, I too Rrp ) s ZE.

Cone rigidity ng,-

- rjigj . Rc -15;] 7 - Cn -Drp'

.

⇒ dah ( Bpp,

( x; ) . Bizrplzfti ) - Elst niv ) Tp R .

( ( Zp ) i une diam 2-p

E IT.

day I Bplol . Bplzp" )

← deal Belo ) . Bplxjsrfg; ) ) t day ( Brlxjirp-ZSI.BIZ ' )

< Ilan ,v ) R

Brlzp't ) → Belo ) . ( p → is ) .

⇒ Y is a metric cone -

• X e RIX), i.e . IR

"

is a tangent cone at x .

⇒ every tangent cone at x -

- IR"

.

Suppose .

( R ",it,o ) = bin ( X . RI 'd× .

x ) .

A→ o

tf [ 7 Ol

da, I Bra ( x ) , Br.

Cont ) < Era. for large a

.

For large j ( depending on x,s .

- - ) ," MJx; → x .

day ( Br.Ix) . Braly ) ) e er.

do, Hl Br. lxjl , Br

.Cont ) c 2 Era

.

keifenberg Them ⇒ t r s rallo ,

dont l Brix;) . Br lo"

, ) s tu ( elm r.

Suppose ( ( IZ) , daz, o ) = bin ( X , Sf'

dy , x ) . Sp → o.

p-so

data I B. hits . B,loni ) E day I B. Iz'T , (Bsplxs . sptdx ) )

Spc ratiot DEH ( (Bsplx) , Sf'd× ) , B , lo

"

) )

- Ilan ) . B , Iz't ) E B , lo

" ) .